If your cat isn't use a litter box, but uses other similar ones *with the same litter/size* the chances are it's because they don't like the location of it.
Try moving the one from the closet out to somewhere else.
The inclosed one is a good idea, some cats don't like to use open litter boxes as they have been attacked by the other cats in the home while their trying to go to the bathroom!
My Asim used to ambush Isha alllll the time, so she formed her own habit for not wanting to use the box, which has been corrected with time.
A GOOD kitty litter in my opinion is Papurr, but not every litter will work the same for every cat, so you just have to try a bit of everything.
Try something small grain so it feels very soft on their toes, try unscented ones, try another pelleted paper/wood litter. Try a litter that has bigger clay bigs, try the ones made out of corn cob and wheat, etc.
If you get desperate you can try using something like Carefresh which is soft, or paper towels/toliet paper ripped up and put in the box.
You'd be amazed what things will work and what might not.
I know a guy who's cat wouldn't use the box at all, but would use the carpeted floor, so he decided to buy a few scraps of carpet for a couple bucks, placed it in her litter box, and she's been faithfull to her box ever since!